Power factor correction drive circuit topologies and control for switched reluctance machines

ABSTRACT

Drive circuits that provide power factor correction and input current waveform shaping for controlling the speed and torque in a switched reluctance machine (SRM). The machine&#39;s phase windings are split into two segments, one of which is used for active power factor correction, input ac current waveform shaping and partial torque generation and the other of which is used for torque generation.

RELATED APPLICATIONS

This application is a Divisional application of U.S. application Ser.No. 13/074,628 filed Mar. 29, 2011, which is a Non-Provisionalapplication of U.S. Provisional Application 61/318,506 filed Mar. 29,2010, the disclosures of which are incorporated herein by reference intheir entireties.

BACKGROUND OF THE RELATED ART

Power factor is defined as the cosine of the phase angle between theinstantaneous alternating current (ac) voltage and current consideringonly the fundamental values of the ac voltages and currents. Unity powerfactor is achieved when the phase angle between the voltage and currentis zero. It is desired to have unity power factor when drawing powerfrom the utility so that only real power is drawn from the utility andno reactive power is drawn from the utility by. The investment on thegenerating and distribution equipments from the utility side will beminimized when only real power, having a power factor of unity, is drawnfrom the utility.

Power converts having rectifiers distort ac current from the utilitysupply, leading to non-sinusoidal current waveforms that introduceharmonics other than the fundamental that are undesirable in theoperation of the utility. Additionally, these harmonics contribute toadditional losses that do not exist when only sinusoidal currents arebeing drawn from the utility. Therefore, power factor correction andinput current waveform shaping are of importance in motor drivesapplications, because of regulations and incentives from utilities toencourage users to build unity power factor operation into their systemsso as to draw sinusoidal current from the utility. Common ways toincorporate these features are: (1) to add a separate unity power factor(UPF) correction circuit, which is an expensive approach and takesadditional space and volume for installation, and (ii) to reuse afull-bridge controlled rectifier operating in a boost mode, which is anexpensive solution.

A solution was patented by Krishnan Ramu (U.S. Pat. No. 7,271,564,Issued: Sep. 18, 2007) which addresses these challenges with a singletransistor for a two phase machine. There are distinct disadvantagesassociated with the single-transistor power converter. One disadvantageis the limited torque generating region for a two phase machine. Whenphase A is conducting, phase B has to conduct, too. The torqueproduction of these two phases are usually of opposite polarity most ofthe time. Therefore, the net torque production in such a circuit withtwo phase windings will have reduced output. A reduced torque outcomecan be proven easily by looking at the instantaneous torque generationin two phases of the SRM.

An exception can be made such that that the currents in the two phasesis unequal, so the torque contributions from the two phases are unequalin magnitude. The net torque is still reduced when the currents in thephases are unequal. Moreover, the reduced or smaller torque is producedin every alternate torque generation region of either phase A or B,whichever can produce the maximum torque compared to the other phase.Assume phase B has less winding turns and phase A has more turns. Theunequal number of turns between the phases makes phase B the auxiliaryphase, with smaller torque generating capability compared to phase A.Therefore, T_(ea)-T_(eb), where T_(ea) and T_(eb) are the torque due tophase A and phase B of the SRM, is positive when phase A's torquegeneration region is positive. This net positive torque comes everyalternate phase cycle thus average torque produced is halved, resultingin smaller torque output.

A circuit with two switches and two diodes per phase requires externalboost inductors for each phase to address the problem of power factorcorrection. The external inductors are expensive, they requireadditional space in the motor drive system, and they require additionalcooling to dissipate core and resistive winding losses. Mostimportantly, these external inductors receive current from the ac sidebut do not produce any useful torque, as they are not part of theelectromagnetic system inside the SRM. The lack of useful torqueprovided by the external inductors is the biggest negative of thecircuit and its operation.

SUMMARY OF THE INVENTION

An object of the invention is to address the above-describedshort-comings of the related art by employing split windings in aswitched reluctance machine (SRM) that form part of the power factorcorrection circuit.

In accordance with the exemplary embodiment of the present invention anSRM is provided that comprises a stator having plurality of poles eachof which has its concentric winding that are connected amongstthemselves in a manner to achieve a required number of machine phasesand a rotor having a plurality of poles with neither windings norpermanent magnets on the rotor poles. An electronic power converter thatis capable of controlling the power flow to the SRM phase windings withone transistor per phase winding that requires split of a phase windingin to two parts, that are not necessarily equal but can be equal ifdesired. These two parts of the phase winding enable power flow throughthe power converter with the transistor and diodes. One part, calledfirst part, of the phase winding allows flow of power from the ac inputsupply with sinusoidal part currents thus contributing to a varyinginput power to a part of the machine output. The other part, calledsecond part, of the same phase winding allows the flow of power from adc capacitor charged previously from the ac supply thus providing aconstant power flow to this part of the winding or it can be modulatedto provide a varying power input to the machine such that the varyinginput power to the first part of the machine phase in combination withthe varying part of the second part of the machine phase can result inconstant power input. An electronic power converter is the object ofinvention which allows for the split phase windings to be connected withonly one transistor per phase to supply power to both parts of the phasewinding and in the process control the input power factor on the acsupply input side. Control strategies to realize the power factorcontrol on the ac supply input side drawing part of full sinusoidalcurrent on the ac supply input side while simultaneously maintaining therequired power and torque output in the SRM is another object of theinvention with the electronic power converter invention. The object ofthe invention is to regulate the air gap power while maintaining thespeed command enforced using air gap power feedback control andsimilarly enforce torque control with torque feedback control with thepower electronic converter operating on a SRM in the invention. Furtherobject of the invention is to make the air gap power control insensitiveto parameter variations such as resistive losses to work with theinvention power electronic converter and the SRM. The power electronicconverter invention is further modified to have only one dc capacitorinstead of two capacitors for compactness and low cost implementationand that is another object of the invention. Split phase windings with atwo transistors per phase configuration in the invention allows foranother object of invention of both power factor control at the input acsupply side and providing for continuous power control of the entirephase winding. Modifications of the invention with one transistor perpart of the phase winding to a total of three transistors for the fourparts phase windings of the two phase SRM are the objects of theinvention to provide a class of power factor correcting SRM drives withdifferent performance characteristics to cater to various applicationsthat may arise in the market. Further object of the invention is anelectronic power converter circuit to work with split phase windings anda small capacitor and only transistor and one diode per phase of themachine winding to provide power factor control at the input ac supplyside, allowing to draw sinusoidal current on the ac supply input sidewhile simultaneously providing torque and power control to the SRMwithout using any external inductors or additional devices to achievethe same. A further object of the invention is to have a SRM withwindings that can be part of the phase windings and that can be placedon the back iron of the stator lamination stack of the SRM. Furtherobject of the invention is that these windings placed on the back ironof the SRM stator will produce flux that can be additive or subtractiveto the fluxes engendered by the winding on the stator poles.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a single switch per phase converter without a fulldiode bridge rectifier for a two phase SRM;

FIG. 2 illustrates the advance commutation of current in a phase windingto prevent negative torque generation;

FIG. 3 illustrates a flow chart for control of the SRM of FIG. 1;

FIG. 4 illustrates phase A currents with both coils having currents;

FIG. 5 illustrates phase A currents and voltages during one PWM cycle;

FIG. 6 illustrates the evaluation of instantaneous duty cycle referencefor phase A for control system 1;

FIG. 7 illustrates schematic of control system 2;

FIG. 8 illustrates gate processor action and resulting waveforms fromcontrol system 2;

FIG. 9 illustrates the current control of phase A;

FIG. 10 illustrates gate signal processor schematic for current controlillustrated in FIG. 9;

FIG. 11 illustrates the implementation block diagram for control system3;

FIG. 12 illustrates the parameter insensitive implementation of controlsystem 3;

FIG. 13 illustrates the implementation in block diagram form of controlsystem 4;

FIG. 14 illustrates an alternative realization of the invented converterin FIG. 1;

FIG. 15 illustrates a circuit with power factor correction for singlephase SRM;

FIG. 16 illustrates a circuit with power factor correction derived fromthe converter in FIG. 15, for a two phase SRM with two part windings ineach phase;

FIG. 17 illustrates a circuit with power factor correction derived fromconverter in FIG. 15 with reduced number of switches for a two phaseSRM;

FIG. 18 illustrates a circuit for power factor correction and drive fora two phase SRM with single switch per phase;

FIG. 19 illustrates a conventional 4/2 SRM with phase windings on theback iron for power factor correction;

FIG. 20 illustrates a 4/2 SRM lamination;

FIG. 21( a) illustrates a method to prevent the windings on the backiron by salient features on the outside of the stator lamination;

FIG. 21( b) illustrates methods to prevent the windings on the back ironby salient features on the inside of the lamination.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 illustrates a single switch per phase power converter 100 for atwo phase SRM. Power converter 100 has one transistor for each phase,each phase comprising two coils. Consider a two phase SRM with phases Aand B having exact equivalency in windings. Each phase has multiplecoils. For illustration consider them having two coils per phase, i.e.A₁ 104 and A₂ 105 for phase A and B₁ 106 and B₂ 107 for phase B. Notethat A₁ 104 and A₂ 105 and likewise B1 106 and B2 107 need not haveequal number of turns but A₁ 104 and B1 106 and A₂ 105 and B₂ 107 areassumed to have equal number of turns to optimize the output torque.Many phase coil arrangement techniques are possible for realizing theSRM. U.S. Provisional Patent Application Ser. No. 60/955,661 by the sameinventor Krishnan Ramu titled Power Factor Correction for SwitchedReluctance Machines describes SRM winding configurations in FIG. 3, FIG.4, and FIG. 6 that are suitable for power factor correction drivecircuit 100. Transistor T_(A) 102 drive phase A and transistor T_(B) 103drives phase B.

Consider the operation of the circuit associated with phase A windingcoils A₁ 104 and A₂ 105. At the start of the circuit operation andimmediately after energization of the circuit, capacitor C_(A) 108 ischarged in the positive cycle of input alternating current (ac) voltage101 and C_(B) 109 is charged to peak of input ac voltage 101 v_(i)during negative half cycle of input ac voltage 101 application.Capacitor C_(A) 108 is charged through diode D₂ 110. Diode D₂ 110ensures that voltage across C_(A) 108, V_(s1), is maintained to somelevel to supply winding A₂ 105 when ac input voltage v_(i) 101 is havinga negative half cycle, with D₁ 111 reversed biased as a result. Thepresence of diode D₂ 110 makes such a difference from previous art inconnection with SRM drive in that it provides a path for energy input toA₂ 105. Coil A₂ 105 is assured of energy input when so desired and whenenergy becomes unavailable from ac supply during negative half-cyclesfor phase A (and likewise for positive half-cycles for phase B overcomeby D₄ 114). The addition of diode D₂ 110 removes one of the majordrawbacks of the power factor correction circuits in U.S. ProvisionalPatent Application Ser. No. 60/955,661. With the charging of capacitorC_(A) 108, voltage V_(s1) can be assumed to be a constant and remains soon average during the operation of the circuit.

Assume voltage v_(ra) is positive and phase A is in a region whenexcited that will produce a positive or motoring torque. Turning on thegate signal of transistor T_(A) 102 will apply a voltage v_(ra) onwinding A₁ 104, enabling a current i_(A1) in it. At the same time, avoltage V_(s1) is applied across coil A₂ 105 initiating a current i_(A2)in it. Transistor T_(A) 102 carries both the currents i_(A1) and i_(A2)and

i _(TA) =i _(A1) +i _(A2)  (1)

The current in A₁ 104 is programmed to be sinusoidal. The sinusoidalcurrent can be realized through either hysteresis or pulse widthmodulation (PWM) control. Considering hysteresis current control forillustration here, transistor T_(A) 102 is turned off when the currentin coil A₁ 104 exceeds its command value (reference value) i*_(A1) bythe hysteresis window Δi.

The switching logic then is,

If (i _(A1) −i* _(A1))≧Δi, turn off T _(A)  (2)

If (i _(A1) −i* _(A1))≦Δi, turn on T _(A)  (3)

Under condition (2), i.e., when T_(A) 102 is turned off, current i_(A1)is diverted through D_(A) 114 to capacitor C_(A) 108, thus charging it.At the same time, the current in coil A₂ 105 is also rerouted through D₂110 and back to A₂ 105 with the result that the applied voltages acrossA₁ 104 and A₂ 105 are:

v _(A1) =v _(ra) −v _(s1)  (4)

v _(A2)=0  (5)

And the currents in transistor T_(A) 102 and diodes D₁ 111 and D_(A) 114when T_(A) 102 is turned off are,

i _(DA)=(i _(A2) +i _(A1))  (6)

i _(D1) =i _(A1)  (7)

i _(TA)=0  (8)

Turning on and off transistor T_(A) 102 goes on in the entire phasecycle when phase A remains in the positive torque region for motoring.Before the phase can generate negative torque, transistor T_(A) 102 isturned off for good. The current in coil A₁ 104 will decay fast and itis determined by the voltage (v_(ra)-V_(s1)) to commutate the current.The current in coil A₂ 105 will take longer time to decay as the currentis freewheeling and zero voltage is applied across it. The substantialnegative voltage in coil A₁ 104 is not applied to A₂ 105. Therefore, inorder not to incur negative torque generation in A₂ 105 (more than in A₁104), the turning off of transistor T_(A) 102 has to be predeterminedand initiated well before the negative torque generation region starts.

FIG. 2 illustrates the advance commutation of current in a phase windingto prevent negative torque generation. Angle θ_(C) is the angle ofadvance commutation before current i_(A2) enters a negative torqueregion. Between θ₁ and θ₂, torque produced will be positive and betweenθ₃ and θ₂, negative torque is produced when a current is present in acoil. The notations in FIG. 2 are:

L_(A) ₂ →Inductance of coil A₂ 105 vs rotor position for a fixed currentin it.

T_(L) _(A2) →Electromagnetic torque generated by A₂ 105 when currenti_(A2) is present in it.

i_(A) ₂ , i*_(A) ₂ →A₂ 105 current and its command.

Consider v_(ra) being negative, i.e., the input ac is in negative halfcycle 101. When v_(ra) is negative, turning on transistor T_(A) 102 willonly energize winding A₂ 105 with the energy stored in capacitor 108 andtorque can be harvested. Note that energy required to charge capacitorC_(A) 108 is always available through diode D₂ 110. Nevertheless,capacitor C_(A) 108 has to be sized to contain energy to support coil A₂105 current during negative half cycle of the ac supply input. Circuitsin U.S. Pat. No. 7,271,564 cannot energize both coils of a phase asenergy in C_(A) 108 is limited by the boost operation only. Hence,capacitor C_(A) 108 would have a smaller energy storage, therebycrimping the torque generation of coil A₂ 105 during negative halfcycles of ac supply input 101 to the circuit.

It should be noted that phase B operates in a similar fashion to phaseA. Transistor T_(B) 103 can draw current from the ac supply 101 toenergize B₁ 106 only when v_(rb) is positive. When v_(rb) is negative,energy stored in capacitor C_(B) 109 is used to energize coil B₂ 107,and diode D₃ 113 is reverse biased. Diode D₄ 112 charges C_(b) 109during the negative half cycle of ac supply 101.

Control of Current in Phase A:

Based on the understanding of circuit 100 in FIG. 1 developed in thedescription above, the control of phase A is derived in the following.The objective of the control system is to get the maximum torque out ofphase with one or both of its coils working. The coils in a phase maywork depending on the positive or negative half cycles of ac supply 101.

FIG. 3 illustrates a flow chart 300 for control of the SRM of FIG. 1.Suppose phase A is selected for activation within decision block 310.Phase A is activated with a small initial duty cycle d₀ by switching onthe appropriate transistor in block 320. The coil currents i_(A1) andi_(A2) are measured according to block 330. If both coil currents aregreater than zero, then use one algorithm to determine the duty cycle asper block 340 based on a torque command. If only coil A₂ 105 has currentbecause the ac input is going through a negative half cycle andi_(A1)=0, then use a different algorithm to determine the duty cycle.After the duty cycle is determined, update the time with the PWM periodT 350 and see whether the phase conduction period is over or not block340. If the conduction period is over, proceed to 370 and change thecontrol of phase B. If phase B's activation has to be advanced, evenbefore phase A's conduction period is over, it can be simply done bycalculating the onset of phase B based on cot and then a branching canbe done. The variable ω is the speed of the motor and variable t is thecurrent time.

FIG. 4 illustrates 400 phase A currents with both coils having currents.The current in coil A₁ 104 is piece-wise sinusoidal 402, and in coil A₂105 it is rectangular with switching current ripples superimposed 403.The current reference for coil A₁ 104 has to follow a sinusoidal shapeto make the ac supply current sinusoidal. The current reference for coilA₂ 105 will be more or less flat and rectangular in shape. The torquereference, T*_(e), is usually apportioned to the coil A₁ 104 and A₂ 105torque reference, viz., T_(e1) and T_(e2), respectively and given by

T* _(e) =T* _(e1) +T* _(e2)  (9)

The key to apportioning torque and calculating the duty cycle oftransistor T_(A) 105 lies in the derivations that follow. The torque ineach coil is found explicitly in terms of the input voltage and dutycycle of transistor 105. Then their references become obvious. Only onePWM period is considered and the corresponding voltages applied to coilsA₁ 104 and A₂ 105 of phase A are found and plotted in FIG. 5 500 {fromequations (4) and (5)}.

In the equations that follow, λ₁ is the flux linkage of A₁ 105, V_(m) isthe peak voltage of ac supply 101, and d is the duty cycle of T_(A) 102.Assuming V_(s1) is a constant, the flux linkage of coil A₁ 104 iswritten as:

$\begin{matrix}\begin{matrix}{{\lambda_{1}( {t + T} )} = {{\int{V_{A\; 1}{t}}} + {\lambda_{1}(t)}}} \\{= {{\int_{t}^{t + {dt}}{V_{m}{\sin ( {\omega \; t} )}\ {t}}} +}} \\{{{\int_{t + {dt}}^{t + T}{{- \lbrack {v_{s\; 1} - {V_{m}{\sin ( {\omega \; t} )}}} \rbrack}\ {t}}} + {\lambda_{1}(0)}}}\end{matrix} & (10) \\{{But}\mspace{14mu} \begin{matrix}{{\int_{t}^{t + {dt}}{V_{m}{\sin ( {\omega \; t} )}\ {t}}} \approx {{dT}\{ \frac{{V_{m}{\sin ( {\omega \; t} )}} + {\sin ( \overset{\_}{{\omega \; t} + {dT}} )}}{2} \}}} \\{\approx {{{dT} \cdot V_{m}}{\sin ( {\omega \; t} )}}}\end{matrix}} & (11)\end{matrix}$

assuming dT<<t

likewise,

∫_(t+dt) ^(t+T)−(v _(s1) −v _(m) sin(ωt))dt=−v _(s1)(1−d)T+(1−d)TV _(m)sin(ω t+dT )≈(1−d)T[V _(m) sin(ωt)−(1−d)V _(s1)]  (12)

Then the flux linkage of coil A₁ 104 are,

λ₁(t+T)=TV _(m) sin(ωt)−T(1−d)V _(s1)+λ₁ t  (13)

where λ₁(t) is the initial condition of λ₁ at time t.

The current in A₁ 104 is calculated:

$\begin{matrix}{i_{A\; 1} = {\frac{\lambda_{1}}{L_{A_{1}}} = {i_{1} = \frac{{{TV}_{m}{\sin ( {\omega \; t} )}} - {{T( {1 - d} )}V_{s\; 1}} + {\lambda_{1}(t)}}{L_{A_{1}}(t)}}}} & (14) \\{{i_{1}( {t + T} )} \approx {\frac{T\{ {{V_{m}{\sin ( {\omega \; t} )}} - {( {1 - d} )V_{s\; 1}}} \}}{L_{A_{1}}(t)} + i_{1}}} & (15)\end{matrix}$

Likewise, the flux linkages in A₂ 105 are:

$\begin{matrix}{{\lambda_{2}( {t + T} )} = {{\int_{t}^{t + {dT}}{V_{S\; 1}\ {t}}} + {\lambda_{2}(t)}}} & (16) \\{{\lambda_{2}( {t + T} )} = {{dTV}_{s\; 1} + {\lambda_{2}(t)}}} & (17) \\{{i_{A_{2}}( {t + T} )} = {{i_{2}( {t + T} )} = {\frac{\lambda_{2}( {t + T} )}{L_{A_{2}}(t)} = {\frac{{dTV}_{s\; 1}}{L_{A_{2}}(t)} + {i_{2}(t)}}}}} & (18)\end{matrix}$

where λ₂ is the flux linkage of A₂ 105

$\begin{matrix}{{{and}\mspace{14mu} {where}\mspace{14mu} {i_{2}(t)}} = \frac{\lambda_{2}(t)}{L_{2}(t)}} & (19)\end{matrix}$

The torques contributed by currents in A₁ 104 and A₂ 105 are:

$\begin{matrix}{T_{e\; 1} = {\frac{1}{2}i_{1}^{2}\frac{{L_{A_{1}}(t)}}{\theta}}} & (20) \\{T_{e\; 2} = {\frac{1}{2}i_{2}^{2}\frac{{L_{A_{2}}(t)}}{\theta}}} & (21)\end{matrix}$

And the total phase A torque is:

$\begin{matrix}{\mspace{79mu} {{T_{e} = {T_{e\; 1} + T_{e\; 2}}}{{T_{e}( {t + T} )} = {{\frac{1}{2}{( \frac{{L_{A_{1}}(t)}}{\theta} )\lbrack {\frac{T\{ {{V_{m}{\sin ( {\omega \; t} )}} - {( {1 - d} )V_{S\; 1}}} \}}{L_{A_{1}}(t)} + {i_{1}(t)}} \rbrack}^{2}} + {\frac{1}{2}{( \frac{{L_{A_{2}}(t)}}{\theta} )\lbrack {\frac{{dTV}_{S\; 1}}{L_{A_{2}}(t)} + {i_{2}(t)}} \rbrack}^{2}}}}}} & (22)\end{matrix}$

FIG. 6 illustrates the evaluation 600 of an instantaneous duty cyclereference for phase A. At the onset of phase A's energization, theinitial conditions are zero for flux linkages and currents. Therefore,i₁(t) and i₂(t) are known as well as the inductance slopes with respectto rotor position and inductance for the given current and position. Thesetting of initial conditions and the determination of inductance slopesand inductances for a given current and positions are illustrated inblock 601. From block 601, it is concluded that the torque is a functionof duty cycle d and time, t, i.e., ωt=θ_(s). Currents i₁(t) and i₂(t)are solved from equation (15) and (18) for every PWM cycle andsubstituted in (22) to yield the solution for torque in block 601.

From the torque expression in (22), reference torque for phase A can bewritten by assuming corresponding variables in reference form as,

$\begin{matrix}{{T_{e}^{*}( {t + T} )} = {{\frac{1}{2}{( \frac{{L_{A_{1}}( {\theta,i} )}}{\theta} )\lbrack {\frac{{{T \cdot V_{m}}{\sin ( {\omega \; t} )}} - {( {1 - d^{*}} )V_{S\; 1}}}{L_{A_{1}}( {\theta,i} )} + {i_{1}(t)}} \rbrack}^{2}} + {\frac{1}{2}{( \frac{{L_{A_{2}}( {\theta,i} )}}{\theta} )\lbrack {\frac{d^{*}{T \cdot V_{S\; 1}}}{L_{A_{2}}( {\theta,i} )} + {i_{2}(t)}} \rbrack}^{2}}}} & (23)\end{matrix}$

where d* is the duty cycle reference at time t. Therefore, given thetorque reference at any time t, the duty cycle can be computed byrearranging (23), shown in 602. After the duty cycle is applied, phaseA's coil currents are read for control during the next PWM cycle 603.The rotor position is updated in 604 and inductance and inductanceslopes are calculated for the next PWM cycle in 605. The duty cycle d*is updated for the next PWM cycle by following the cyclical path through606, leading to block 602.

The above system for control of the SRM with novel converter can betermed control system 1. Other control strategies are derived leading toeasier computation.

Control System 2:

The parameter dependencies of control system 1 lead to computationallyintensive implementations. Control system 1 may not be suitable for lowcost applications. Simpler applications such as fans, require acomputationally light control scheme.

FIG. 7 illustrates schematic 700 of control system 2. The torque commandis used to derive an equivalent current command 701 with a proportionalfactor and square rooting function as 702:

I*=(√{square root over (T* _(e))})k _(it)  (24)

The factor K_(it) can be formed from an experimental or a simulationbased torque versus current relationship. The current command I* 701 isa dc value illustrated by 703. The current reference is then convertedinto an equivalent sine reference i_(s)* 704 by multiplying it with anattenuated rectified signal derived from the supply ac input voltage.The attenuated ac signal is sine reference 705. The shape of theequivalent sine reference is illustrated in 706. The rectified sinevoltage is obtained from the main circuit variable v_(ra), also. Thegating signal for the transistor is developed from i_(s)* 704 bymultiplying it with the phase enabling signal for the correspondingphase A or B. Likewise, it then can be turned off after a dwell angle,which corresponds to a dwell time of conduction for either phase givenas:

$\begin{matrix}{T_{d\omega} = \frac{\theta_{\omega}}{\omega_{m}}} & (25)\end{matrix}$

where T_(dω) is the dwell time of conduction, θ_(dω) is the dwell anglein radians (rad), and ω_(m) is the rotor speed in rad/sec. θ_(dω) is aninput variable.

FIG. 8 illustrates gate processor action and resulting waveforms fromcontrol system 2. The gating signal for phase A or B is generated in thegating process and illustrated in FIG. 8 800. The phase conduction(dwell) pulses for phases A and B are generated and conduction (dwell)pulses merge with i_(s)* for either time duration. A_(dω) is theconduction pulse for phase A in 800. Then current command i_(s)* duringthat dwell interval serves to put and enforce a current in either phase.

The phase conduction phase A_(dω) can be generated from the rotorposition at which the phase starts generating positive (i.e., motoring)torque through all positions in which it can keep generating the torque,which is given by the dwell angle. The dwell angle can be an externalinput and can be made a function of speed and, to an extent, torquerequest also. The phase current command i*_(A) can be generated bymultiplying A_(dω) and i*_(s) and normalizing it, as shown in 801.Current i*_(A) is a small segment of the sine waveform i*_(r) alongwhich the phase A current has to be generated. Currents generated fromphase current command i*_(A) 801 are generated by the phase coils A₁ 104and A₂ 105 in graphs 802, and 803, respectively. The duty cycle forphase A is obtained by the closed loop current control as shown in FIG.9 900. The feedback current 902 in coil A₁ 104 is filtered in 908 andcompared to its command i*_(A) 901 and the error 903 calculated through904 is amplified and conditioned by a current controller of aproportional plus integral (PI) type 905. The output of currentcontroller 905 is the duty cycle reference for transistor T_(A) 102 anddenoted by d* 906. The duty cycle command 906 is fed to converter 907 todrive SRM 909.

In a schematic form, the gate signal processor has the form 1000illustrated in FIG. 10. A gate signal processor 1010 takes the phase Aturn-on signal, based on whether phase A will generate positive torqueor not, and the dwell angle to generate phase A's normalized conductionpulse A_(dω) in a conduction pulse generator 1011. Conduction pulseA_(dω) is multiplied with the current command i_(s)* by a multiplier1012. The output of multiplier 1012 is phase A current command i_(A)*.The phase current error from 1013 for coil A₁ 104 is fed to currentcontroller 1014, whose output is duty cycle d* for phase A.

In control system 2, coil A₁ 104's current is regulated and the dutycycle determined for coil A₂ 104 current is applied equally to generatethe current in coil A₂ 105. Therefore, coil A₂ 105 current control isslave to current control in coil A₁ 104. Whether the combined currentswill produce the desired torque will be determined by the outer speedfeedback control loop. The speed error will determine the torquecommand. Accordingly, the currents in phase A will be coordinated toyield the desired torque. The speed error can be sampled once in a phaseconduction period so that the fluctuation in input current can beminimized Control system 2 does not involve any motor parameters such as

$L_{A_{1}},\frac{L_{A_{1}}}{\theta}$

and is insensitive to variations in phase resistances. Hence, controlsystem 2 is easy to implement and easier to tune in the field. Also, theamount of computations for control system 2 is minimized making itattractive to use low cost processors to implement control system 2.

Control System 3:

Control system 3 neither depends on the parameters of the machine nor onthe closed-loop feedback control of both phase winding currents. Theimplementation schematic of control system 3 1100 for one of the twophases is shown in FIG. 11. Control system 3 is based on the fact thatcommanded air gap power will generate an air gap power to match it.Airgap power control is enforced by a feedback control of speed andresulting air gap power feedback control. The air gap power feedbackcontrol forms the inner loop and speed feedback control forms the outercontrol loop. The speed control loop has a subtractor 1120 to subtractthe actual machine speed from its reference to generate a speed error1121. Speed error 1121 is amplified, conditioned, and normalized using aPI type of controller 1122 whose output is a torque command T*_(e) 1123.An air gap power command 1124 is generated by a multiplication 1125 ofspeed and torque commands.

The air gap power generated in the machine is found by the instantaneouspower in each phase of the machine at a given time. It is arrived at byfinding the product of current and voltage in each coil of a phase andthen subtracting the resistive losses from it. To estimate phase airgappower for phase A 1110, phase voltages and currents from each coil ofphase A are sampled and multiplied through multipliers 1111, 1112 toobtain the input power to both coils of phase A. Resistive losses areestimated by first taking the square of the current in each phase coilthrough multipliers 1113 and 1114. The squares of the coil currentsaren't are scaled by their respective coil resistances in 1115 and 1116to obtain the resistive loss in each coil of the phase. The resistivelosses are subtracted from the input power to their respective phasecoils through subtractors 1117 and 1118 to give the airgap powercontributed from each phase coil. These airgap powers are summed in 1119to give the total airgap power of phase A.

The difference 1126 between the airgap power command and the airgappower is an airgap power error 1127. Airgap power error 1127 is fed intoan airgap power controller 1128 whose output is a duty cycle command1129, which is fed to the power converter.

Nomenclature for FIG. 11:

ω*_(m)—Speed command

ω_(m)—Speed

Δω_(m)—Speed error→Δ(ω*_(m)−ω_(m))

T*_(e)—Torque command

Symbol ‘X’—Multiplier block

P*_(a)—Airgap power command of

P_(a)—Airgap power

ΔP_(a)—Airgap power error

v_(A) ₁ , v_(A) ₂ —voltages across coils A₁ 104 and A₂ 105, respectively

i_(A) ₁ , i_(A) ₂ —current in coils A₁ and A₂, respectively

R_(A) ₁ , R_(A) ₂ —resistance of coils A₁ 104 and A₂ 105, respectively

P_(A) ₁ —Air gap power of coil A₁ 104

P_(A) ₂ —Air gap power of coil A₂ 105

d*—Duty cycle command

For an n-phase machine, note that ‘n’ number of phase airgap powerestimators have to be incorporated in the block diagram. If more thanone phase is contributing to torque generation at an instant, then P_(a)consists of the air gap powers of all the phases whose phase currentsare greater than zero. These are the only modifications to make tocontrol block diagram in 1100 to make it applicable to the control of ann-phase machine control. A duty cycle 1129 command that emerges out ofthe airgap power controller 1128 is steered to various gating circuitsof phase transistors using phase conduction periods of correspondingphases. The steering of duty cycle 1129 to various gating circuits isderived in gate signal processor schematic in FIG. 10. Usually theoutgoing phase is not energized and only the incoming phase is energizedin a multiphase SRM. If the conduction of both phases overlap, it is fora small time and during that small time the same duty cycle is passed onto the transistors of the two phases.

To start and run the SRM without compensating for stator resistivelosses, a feed-forward signal from the torque command can be utilized.When the feed-forward signal from the torque command is utilized, allthe computations of stator resistive losses go away making theimplementation simpler. Such an implementation is shown in FIG. 12 1200.

The stator resistive losses can be predetermined from simulation for agiven torque request and the predetermined losses can be programmed in alook-up table 1201. The applicable loss can be recalled and addedthrough summer 1202 with the air gap power command giving the commandedinput power. The actual input power of the machine is subtracted bysummer 1202 from the commanded input power to provide an error in inputpower which can be processed through a PI type controller, known asinput power controller 1203. The actual input power to the machine isestimated by calculating the input power to each coil in a phase throughmultipliers 1204, 1205. The input power to each coil is summed through asummer 1206 and then conditioned through a filter 1207 to get the totalphase input power. The feedback control implements the input powercontrol as against the airgap power. Control implement in FIG. 12 1200is free of parameters of the machine.

Implementations for control strategies 2 and 3 require sensing of phasevoltages (i.e. coil voltages for each phase). It can easily be doneusing resistors across the coils and only part of it used for signalextraction. An alternative control system not using voltage signals isdeveloped in the following section.

Control System 4:

Control System 4 compares a command torque and a computed torque usingall currents of coils 104, 105, 106, 107 and the rotor position in atorque estimator 1320 to develop a torque error 1302. The torque errorthen is processed through a PI type controller, knows as a torquecontroller 1303. The proposed system is principally a feedback controlof torque. Therefore, it will be faster and capable of excellentresponse. The machine torque from all phase coil currents can becomputed and stored in tables 1304, 1305, 1306, 1307, which controltorque due to currents in each coil of a phase. Tables 1304, 1305, 1306,1307 each contain data of torque versus a phase coil's current and rotorposition. Therefore, given a specified coil, a current and a rotorposition as inputs, the table can output the torque in the specifiedphase coil. Tables 1304, 1305, 1306, 1307 can be developed from machinecharacteristics obtained either from experimental results (preferredapproach) or from finite element based analysis. Then, theimplementation is independent of explicit machine parameters and onlydepends on machine currents, thus simplifying the implementation.

FIG. 13 illustrates the implementation 1300 of control system 4 in blockdiagram form. Phase A torque is the sum 1308 of the torques produced bycoils A₁ 104 and A₂ 105, and similarly phase B torque is the sum 1309 ofthe torques produced by coils B₁ 106 and B₂ 107. The sum of the torquesproduced by phases A and B constitutes the air gap torque of the machine1310 at a given instant, which is then compared with the torque commandin a summer 1301. The torque command in a speed controlled drive systemcan be derived from speed error as shown in 1000, 1100, 1200. The torqueestimator can be easily realized, based on the schematic, using adigital signal processor. The duty cycle command there is implementedfor each PWM cycle as discussed elsewhere.

FIG. 14 illustrates an alternative realization 1400 of the converterillustrated in FIG. 1. In system 1400, only one capacitor 1401 isrequired opposed to two in system 100. Coils A₁ 104 and B₁ 106 can beactivated regardless of whether ac input voltage 101 is going through apositive or negative half cycle. In system 100, A₁ 104 and B₁ 106 couldbe activated in positive and negative half cycles of supply input 101.All the emitters of transistors 1402, 1403 are commonly tied to rail1404, resulting in no isolation required from their gate drives and,hence, in the elimination of one isolated power supply. A diode D5 1405allows capacitor 1401 to be charged by the ac source 101 when itsvoltage is lower than the instantaneous magnitude of ac supply 101.

An alternate realization of the circuit in FIG. 14 is the circuit 1500in FIG. 15 for a single phase SRM with a two part winding. The two coilsof the single phase SRM's phase winding are coil A₁ 104 and coil A₂ 105.Only one capacitor 1501 is required to implement circuit 1500'srealization. Diodes 111, 112, 113, 114 arranged in circuit 1400 form asingle phase rectifier, which is rectifier 1506 in circuit 1500. Themerits of the converter 1500 are: (i) it is capable of power factorcorrection; (ii) it can draw sinusoidal ac input current by switching onand off transistor T_(A1) 1504; (iii) no external inductor is requiredfor power factor correction; (iv) an inductor for boost operation comesfrom part of the phase winding, 104, resulting in: (a) compactness and(b) use of inductor 104 to produce torque while serving to boost anddraw sinusoidal ac input current for power factor correction; and (v)part of the phase winding, i.e., coil A₂ 105, is used by turning ontransistor T_(A2) 1503 to use the energy stored in capacitor 1501,resulting in more torque generation. Freewheeling of i_(A2) is achievedby turning off transistor T_(A1) 1502 (TA₂ 1503 on) or vice versa.Turning off both transistors T_(A1) 1502 and T_(A2) 1503 applies −V_(dc)across A2 105, resulting in faster decay of current i_(A2).

Various control strategies are possible with circuit 1500. Circuit 1500can be modified for a two phase SRM. By duplicating transistors 1502,1503 and diodes 1504, 1505, which are connected to the phase A coils incircuit 1500, a second phase can be connected to the converter to drivea two phase SRM.

FIG. 16 illustrates a circuit 1600 with power factor correction derivedfrom the converter in FIG. 15, for a two phase SRM with two partwindings in each phase. In FIG. 16, transistors 1601, 1602 and diodes1603, 1604 drive phase B coils in the same manner as phase A coils aredriven in circuit 1500. Independent control of phase A and phase Bcurrents are possible with circuit 1600.

FIG. 17 illustrates a two-phase SRM circuit 1700 with power factorcorrection and a reduced number of switches. Circuit 1700 is derived byreplacing transistors 1503, 1601 in circuit 1600 with one transistor1701. Only three transistors are required to drive a three-phase SRM andperform power factor correction. The demerit of using reduced transistorcircuit 1700 is that independent control of currents in A₂ 105 and B₂107 is compromised to a degree, compared to circuit 1600. The demerit ofusing reduced transistor circuit 1700 can be overcome by turning offcurrent in coil A₂ 105 and then energizing coil B₂ 107. Overcoming thedemerit of using reduced transistor circuit 1700 does not preventcontrol of currents in coils A₁ 104 and B₁ 106 to be independent of eachother.

FIG. 18 illustrates a circuit 1800 for power factor correction and driveof a two-phase SRM with a single switch per phase. Transistor 1801drives coil A₁ 104 and transistor 1802 drives coil B₁ 106. Coils A₂ 105and B₂ 107 are energized by energy stored in the two small capacitors C₁1803 and C₂ 1804, respectively. The energy stored in coils 104, 106 isused to boost capacitors 1803, 1804 respectively. Coils A₁ 104 and B₁106 are opposed by voltages in C₁ 1803 and C₂ 1806, respectively, thatturn off current in the coils resulting in faster decay of currents inthem. Limited energy in C₁ 1803 and C₂ 1804 allows for faster decay ofcurrents in A₂ 105 and B₂ 107, respectively. Transistors T_(A) 1801 andT_(B) 1802 have common emitters, thus removing the need for an isolatedpower supply for their gate drives. The currents in A₁ 104 and B₁ 106can be sensed inexpensively with a sensing resistor placed between thetransistor emitter and the common of the dc rectified terminals withvoltage, v_(r).

An SRM with split coils in the phase windings to be used for powerfactor correction has been discussed extensively. Many novelarrangements of windings coils are possible to enable employment of oneof power factor correction circuits in FIGS. 1 and 14-18. Someapplicable SRM winding configurations are presented in FIGS. 3, 4 and 6of U.S. Provisional Patent Application Ser. No. 60/955,661 by the sameinventor Krishnan Ramu titled Power Factor Correction for SwitchedReluctance Machines. U.S. Provisional Patent Application Ser. No.60/955,661 demonstrated putting coils on the back iron of a machine withno flux reversals in any segment of the back iron.

A conventional two phase SRM that has a flux reversal in the back ironcan be modified by the technique illustrated FIGS. 19 and 20. A twophase 4/2 SRM has been chosen for description and can be extended to anynumber of stator and rotor pole combinations. The stator of a 4/2 SRMhas four salient poles, and the rotor has two salient poles. A two-phase4/2 SRM has windings placed on the back iron where the flux does notreverse. The phase winding coils are arranged such that these windingsare contained within the outer dimensions of the stator laminations.FIG. 19 shows the stator lamination with phase windings 1920, 1921,1922, and 1923.

A machine 1900 has triangular sections (1,2,3) 1901, (4,5,6) 1902,(7,8,9) 1903, (10,11,12) 1904 in the stator lamination. For simplerpunching, the corners of stator laminations may be a little rounded.Triangles 1901, 1902, 1903, 1904 are punched out from the outer edge ofstator lamination 1910 to make room for windings to sit on.

For the winding to sit on the inside, correspondingly triangles 1905,1906, 1907, 1908 are added and made to be straight-lined for the windingbase to sit on. Then, the winding is designed to fit in areas 1902,1903, 1904 assuming those sections of back iron do not see a fluxreversal. When windings sit in areas 1902, 1903, 1904, triangle areas1905, 1906, 1907, 1908 are not utilized. Iron is added to triangle areas1905, 1906, 1907, 1908 for machine symmetry. Therefore, the finalmachine 2000 with back iron windings and stator lamination is shown inFIG. 20.

L₁ and L₂ are arranged in such a way that flux produced by them are inthe same direction as the flux that would be made by coils A₁₁ 1920 andA₁₂ 1921 and coils B₁₁ 1922 and B₁₂ 1923. For the excitation shown forphase A (with windings A₁₁ 1920 and A₂₂ 1921) and for phase B (withwindings B₁ 1922 and B₂ 1923), the flux in back iron portions betweenA₁₁ 1920 and B₁₁ 1922 is clockwise and between A₁₂ 1921 and B₁₂ 1923 iscounterclockwise. Coils L₁ 2010 and L₂ 2011 are fixed in the cornerswithout exceeding the outer square (or rectangular) periphery of thestator lamination 1910, to ensure no additional space is occupied by thestator coils.

For coils L₁ 2010 and L₂ 2011 not to slide or move, many arrangementsare possible, such as tying or gluing them. The coil arrangements 2110and 2120 shown in FIG. 21 leave salient side iron at the bottom or topof the stator.

What is claimed is:
 1. A power converter for supplying power to a motor,the power converter comprising for a phase of the motor: no more thantwo diodes; a single capacitor; and no more than two switches that eachconducts current when activated and does not conduct current whendeactivated, wherein: the two diodes, capacitor, and two switches areinterconnected such that when interconnected with a single-phasealternating current supply through a full-wave bridge rectifier andfirst and second series-connected windings for the motor phase: a firstoperational mode of the power converter exists in which current isconducted through a first of the two switches and the first motor phasewinding, a second operational mode of the power converter exists inwhich current is not conducted through the first switch and energy isstored in the capacitor, a third operational mode exists in which energystored in the capacitor during the second operational mode is dischargedby the conduction of current through both of the switches and the secondmotor phase winding.
 2. A power converter for supplying power to amotor, the power converter comprising for two phases of the motor: nomore than four diodes; a single capacitor; and no more than fourswitches that each conducts current when activated and does not conductcurrent when deactivated, wherein: the four diodes, capacitor, and fourswitches are interconnected such that when interconnected with asingle-phase alternating current supply through a full-wave bridgerectifier, first and second series-connected windings of a first of themotor phases, and third and fourth series-connected windings of a secondof the motor phases: a first operational mode of the power converterexists in which current is conducted through a first of the fourswitches and the first motor phase winding, but no current flows througha second of the four switches, a second operational mode of the powerconverter exists in which current is conducted through the second switchand the second motor phase winding, but no current is conducted throughthe first switch, energy is stored in the capacitor during the first andsecond operational modes by: (1) the conduction of current through thefirst motor phase winding and a first of the four diodes to thecapacitor or (2) the conduction of current through the second motorphase winding and a second of the four diodes to the capacitor, a thirdoperational mode exists in which energy stored in the capacitor duringthe first or second operational modes is discharged through the thirdmotor phase winding when current is conducted through the first switchand a third of the four switches, and a fourth operational mode existsin which energy stored in the capacitor during the first or secondoperational modes is discharged through the fourth motor phase windingwhen current is conducted through a fourth of the four switches and thesecond switch.
 3. A power converter for supplying power to a motor, thepower converter comprising for two phases of the motor: no more thanthree diodes; a single capacitor; and no more than three switches thateach conducts current when activated and does not conduct current whendeactivated, wherein: the three diodes, capacitor, and three switchesare interconnected such that when interconnected with a single-phasealternating current supply through a full-wave bridge rectifier, firstand second series-connected windings for a first of the two motorphases, and third and fourth series-connected phase windings for asecond of the two motor phases: a first operational mode of the powerconverter exists in which current is conducted through a first of thethree switches and the first motor phase winding, but no current isconducted through a second of the three switches, a second operationalmode of the power converter exists in which current is conducted throughthe second switch and the second motor phase winding, but no current isconducted through the first switch, energy is stored in the capacitorduring the first and second operational modes by: (1) the conduction ofcurrent through the first motor phase winding and a first of the threediodes to the capacitor or (2) the conduction of current through thesecond motor phase winding and a second of the three diodes to thecapacitor, a third operational mode exists in which energy stored in thecapacitor during the first or second operational modes is discharged bythe conduction of current through the third motor phase winding, thefirst switch, and a third of the three switches, and a fourthoperational mode exists in which energy stored in the capacitor duringthe first or second operational modes is discharged by the conduction ofcurrent through the fourth motor phase winding and the second and thirdswitches.
 4. A power converter for supplying power to a motor, the powerconverter comprising for a phase of the motor: a single diode; a singlecapacitor; and a single switch that conducts current when activated anddoes not conduct current when deactivated, wherein: the diode,capacitor, and switch are interconnected such that when interconnectedwith a single-phase alternating current supply through a full-wavebridge rectifier and first and second windings for the motor phase: afirst operational mode of the power converter exists in which current isconducted through the switch and the first motor phase winding, a secondoperational mode of the power converter exists in which energy stored inthe first motor phase winding is discharged, by the conduction ofcurrent through the first motor phase winding and the diode to thecapacitor, to be stored as energy in the capacitor, a third operationalmode exists in which energy stored in the capacitor during the secondoperational mode is discharged by the conduction of current through thesecond motor phase winding.
 5. A power converter for supplying power tofirst and second series-connected windings of a motor phase, the powerconverter comprising: first and second unidirectional current elementsthat each conducts current in only one direction between a current inputterminal and a current output terminal; a capacitive element for storingenergy across first and second terminals; first and second switches thateach conducts current between a current input terminal and a currentoutput terminal when activated and does not conduct current between thecurrent input and output terminals when deactivated, wherein: the firstterminal of the capacitive element, the current input terminal of thesecond switch, and the current output terminal of the firstunidirectional current element are directly connected by a first node,the current input terminal of the first unidirectional current elementis directly connected to the current input terminal of the first switchby a second node, the current output terminal of the second switch isdirectly connected to the current output terminal of the secondunidirectional current element by a third node, the second terminal ofthe capacitive element is directly connected to the current outputterminal of the first switch and the current input terminal of thesecond unidirectional current element by a fourth node, when: (1) asingle-phase alternating current supply is interconnected with the powerconverter through a full-wave bridge rectifier whose negative voltageterminal is directly connected to the fourth node, (2) the first motorphase winding has a first terminal directly connected to the positiveterminal of the full-wave bridge rectifier and a second terminaldirectly connected to the second node, and (3) the second motor phasewinding has a first terminal directly connected to the second node and asecond terminal directly connected to the third node, then: a firstoperational mode exists in which current is conducted through the firstmotor phase winding, while the first switch conducts is activated, asecond operational mode exists in which current is conducted through thefirst motor phase winding and the first unidirectional current elementto charge the capacitive element, while the first switch is deactivated,a third operational mode exists in which energy stored by the capacitiveelement during the second operational mode is discharged by theconduction of current through the second motor phase winding when boththe first and second switches are activated.
 6. A power converter forsupplying power to first and second series-connected windings of a firstmotor phase and third and fourth series-connected windings of a secondmotor phase, the power converter comprising: first, second, third, andfourth unidirectional current elements that each conducts current inonly one direction between a current input terminal and a current outputterminal; a capacitive element for storing energy across first andsecond terminals; first and second switches that each conducts currentbetween a current input terminal and a current output terminal whenactivated and does not conduct current between the current input andoutput terminals when deactivated, wherein: the first terminal of thecapacitive element, the current output terminals of the first and thirdunidirectional current elements, and the current input terminals of thefirst and third switches are directly connected by a first node, thesecond terminal of the capacitive element, the current input terminalsof the second and fourth unidirectional current elements, and thecurrent output terminals of the second and fourth switches are directlyconnected by a second node, the current output terminal of the firstswitch is directly connected to the current output terminal of thesecond unidirectional current element by a third node, the currentoutput terminal of the third switch is directly connected to the currentoutput terminal of the fourth unidirectional current element by a fourthnode, the current input terminal of the second switch is directlyconnected to the current input terminal of the first unidirectionalcurrent element by a fifth node, the current input terminal of thefourth switch is directly connected to the current input terminal of thethird unidirectional current element by a sixth node, when: (1) asingle-phase alternating current supply is interconnected with the powerconverter through a full-wave bridge rectifier whose negative voltageterminal is directly connected to the second node, (2) the first motorwinding has a first terminal directly connected to the positive terminalof the full-wave bridge rectifier and a second terminal directlyconnected to the fifth node, (3) the third motor winding has a firstterminal directly connected to the positive terminal of the full-wavebridge rectifier and a second terminal directly connected to the sixthnode, (4) the second motor winding has a first terminal directlyconnected to the third node and a second terminal directly connected tothe fifth node, and (5) the fourth motor winding has a first terminaldirectly connected to the fourth node and a second terminal directlyconnected to the sixth node, then: a first operational mode exists inwhich current is conducted through the first motor winding, while thesecond switch is activated, a second operational mode exists in whichcurrent is conducted through the third motor winding, while the fourthswitch is activated, a third operational mode exists in which current isconducted through the first motor winding and the first unidirectionalcurrent element to the capacitive element to charge the capacitiveelement, while the second switch is deactivated, a fourth operationalmode exists in which current is conducted through the third motorwinding and the third unidirectional current element to the capacitiveelement to charge the capacitive element, while the fourth switch isdeactivated, a fifth operational mode exists in which energy stored bythe capacitive element during the third or fourth operational modes isdischarged by the conduction of current through the second motor phasewinding when both the first and second switches are activated, and asixth operational mode exists in which energy stored by the capacitiveelement during the third or fourth operational modes is discharged bythe conduction of current through the fourth motor phase winding whenboth the third and fourth switches are activated.
 7. A power converterfor supplying power to first and second series-connected windings of afirst motor phase and third and fourth series-connected windings of asecond motor phase, the power converter comprising: first, second, andthird unidirectional current elements that each conducts current in onlyone direction between a current input terminal and a current outputterminal; a capacitive element for storing energy across first andsecond terminals; first, second, and third switches that each conductscurrent between a current input terminal and a current output terminalwhen activated and does not conduct current between the current inputand output terminals when deactivated, wherein: the first terminal ofthe capacitive element, the current output terminals of the first andthird unidirectional current elements, and the current input terminalsof the first switch are directly connected by a first node, the secondterminal of the capacitive element, the current input terminal of thesecond unidirectional current element, and the current output terminalsof the second and third switches are directly connected by a secondnode, the current output terminal of the first switch is directlyconnected to the current output terminal of the second unidirectionalcurrent element by a third node, the current input terminal of thesecond switch is directly connected to the current input terminal of thefirst unidirectional current element by a fourth node, the current inputterminal of the third switch is directly connected to the current inputterminal of the third unidirectional current element by a fifth node,when: (1) a single-phase alternating current supply is interconnectedwith the power converter through a full-wave bridge rectifier whosenegative voltage terminal is directly connected to the second node, (2)the first motor winding has a first terminal directly connected to thepositive terminal of the full-wave bridge rectifier and a secondterminal directly connected to the fourth node, (3) the third motorwinding has a first terminal directly connected to the positive terminalof the full-wave bridge rectifier and a second terminal directlyconnected to the fifth node, (4) the second motor winding has a firstterminal directly connected to the third node and a second terminaldirectly connected to the fourth node, and (5) the fourth motor windinghas a first terminal directly connected to the third node and a secondterminal directly connected to the fifth node, then: a first operationalmode exists in which current is conducted through the first motorwinding, while the second switch is activated, a second operational modeexists in which current is conducted through the third motor winding,while the fourth switch is activated, a third operational mode exists inwhich current is conducted through the first motor winding and the firstunidirectional current element to the capacitive element to charge thecapacitive element, while the second switch is deactivated, a fourthoperational mode exists in which current is conducted through the thirdmotor winding and the third unidirectional current element to thecapacitive element to charge the capacitive element, while the thirdswitch is deactivated, a fifth operational mode exists in which energystored by the capacitive element during the third or fourth operationalmodes is discharged by the conduction of current through the secondmotor phase winding when both the first and second switches areactivated, and a sixth operational mode exists in which energy stored bythe capacitive element during the third or fourth operational modes isdischarged by the conduction of current through the fourth motor phasewinding when both the first and third switches are activated.
 8. A powerconverter for supplying power to first and second windings of a motorphase, the power converter comprising: a unidirectional current elementthat conducts current in only one direction between a current inputterminal and a current output terminal; a capacitive element for storingenergy across first and second terminals; a switch that conducts currentbetween a current input terminal and a current output terminal whenactivated and does not conduct current between the current input andoutput terminals when deactivated, wherein: the first terminal of thecapacitive element is directly connected to the current output terminalof the unidirectional current element by a first node, the current inputterminal of the unidirectional current element is directly connected tothe current input terminal of the switch by a second node, when: (1) asingle-phase alternating current supply is interconnected with the powerconverter through a full-wave bridge rectifier whose negative voltageterminal is directly connected to the current output terminal of theswitch and whose positive terminal is directly connected to the secondterminal of the capacitive element by a third node, (2) the first motorphase winding has a first terminal directly connected to the third nodeand a second terminal directly connected to the second node, and (3) thesecond motor phase winding has a first terminal directly connected tothe first node and a second terminal directly connected to the thirdnode, then: a first operational mode exists in which current flowsthrough the first motor phase winding, while the first switch isactivated, a second operational mode exists in which energy stored inthe first motor phase winding is discharged by the conduction of currentthrough the unidirectional current element into the capacitive element,to be stored as energy, and the second motor phase winding when thefirst switch is deactivated, and a third operational mode exists inwhich energy stored in the capacitive element during the secondoperational mode is discharged by the conduction of current through thesecond motor phase winding, while the switch is activated ordeactivated.
 9. A stator for a motor, the stator comprising: multiplestator poles that each has a coil winding around it to induce a fluxthrough the pole when an electrical current flows through the windingand a rotor is aligned to circulate the flux; a coil winding around theback iron of the stator that is disposed at a location of the back ironthat does not experience flux reversal due to the excitation of the coilwindings around the stator poles.
 10. The stator of claim 9, wherein thecoil winding around the back iron produces flux in the same directionthrough the back iron as does the coil windings around two of the statorpoles, when each of the coil windings is excited to generate a flux. 11.The stator of claim 9, further comprising: another coil winding aroundthe back iron of the stator that is disposed at a location of the backiron that does not experience flux reversal due to the excitation of thecoil windings around the stator poles, wherein: the two coil windingsaround the back iron generate flux in opposite directions through theback iron when excited to generate a flux.
 12. The stator of claim 11,wherein each of the two coils around the back iron are disposed atcorners of the back iron.
 13. The stator of claim 12, wherein the twocoils are disposed at opposite corners of the back iron.
 14. The statorof claim 9, further comprising salient portions of the back iron thatpartially enclose the side edges of the iron back winding so as toinhibit the iron back winding from sliding along the back iron betweenstator poles.
 15. The stator of claim 14, wherein the salient portionsare disposed on the inner face of the iron back.
 16. The stator of claim14, wherein the salient portions are disposed on the outer face of theiron back.
 17. The stator of claim 13 wherein the number of stator polesis four.
 18. A method of calculating a duty cycle command of a motorphase having two series-connected coil windings, the method comprising:calculating, with a processor, the duty cycle command by using theequation:${T_{e}^{*}( {t + T} )} = {{\frac{1}{2}{( \frac{{L_{A_{1}}( {\theta,i} )}}{\theta} )\lbrack {\frac{{{T \cdot V_{m}}{\sin ( {\omega \; t} )}} - {( {1 - d^{*}} )V_{S\; 1}}}{L_{A_{1}}( {\theta,i} )} + {i_{1}(t)}} \rbrack}^{2}} + {\frac{1}{2}{( \frac{{L_{A_{2}}( {\theta,i} )}}{\theta} )\lbrack {\frac{d^{*}{T \cdot V_{S\; 1}}}{L_{A_{2}}( {\theta,i} )} + {i_{2}(t)}} \rbrack}^{2}}}$and known values of${T_{e}^{*}( {t + T} )},{\frac{{L_{A_{1}}( {\theta,i} )}}{\theta}\frac{{L_{A_{21}}( {\theta,i} )}}{\theta}},T,V_{m},\omega,t,V_{s\; 1},{i_{1}(t)},{L_{A\; 1}( {\theta,i} )},$L_(A2)(θ, i), and i₂(t), where: t is an initial time, T is a period oftime subsequent to time t, T_(e)*(t+T) is reference torque at time(t+T), θ is a position of a rotor of the motor, L_(A1)(θ, i) is theinductance of a motor coil A₁ at rotor position θ for a fixed current i,L_(A2)(θ, i) is the inductance of a motor coil A₂ at rotor position θfor a fixed current i, ω is the speed of the rotor, V_(m) is the peakvoltage of an alternating current supply, i₁(t) is the inductance slopewith respect to rotor position for a first coil winding of the motorphase, i₂(t) is the inductance slope with respect to rotor position fora second coil winding of the motor phase, and V_(s1) is the voltageapplied across the second coil winding of the motor phase, monitoringthe values of i₁(t) and i₂(t) after a period of time T; andrecalculating the duty cycle command.
 19. A method of generating asignal to control energization of a motor phase, the method comprising:generating a current command from a torque command; generating anequivalent sine reference by multiplying the generated current commandby an attenuated rectified signal obtained from a supply alternatingcurrent (ac) voltage; generating the signal to control energization ofthe motor phase by multiplying the generated equivalent sine referenceby a phase enabling signal for the motor phase.
 20. A method ofgenerating a duty cycle command of a motor phase, the method comprising:generating a normalized conduction pulse based on a signal forenergizing the motor phase and the dwell angle of the motor phase;generating, with a processor, a current command by multiplying thegenerated conduction pulse by a current command for the motor phase;generating a phase current error by subtracting an indicator of thecurrent within the motor phase; and generating the duty cycle commandfrom the generated phase current error.
 21. A method of generating aduty cycle command of a motor phase, the method comprising: estimating,with a processor, phase air gap power from current and voltage appliedto each of two series-connected coil windings of the motor phase;generating a speed error of a rotor of the motor by subtracting therotor's actual speed from its reference speed; generating a torquecommand from the generated speed error; generating an air gap powercommand by multiplying the generated torque command by the rotor'sreference speed; generating an air gap error by subtracting theestimated air gap power from the generated air gap power command; andgenerating the duty cycle command from the air gap error.
 22. A methodof generating a duty cycle command of a motor phase, the methodcomprising: calculating, with a processor, air gap power for the motorphase from current and voltage applied to each of two series-connectedcoil windings of the motor phase; generating a speed error of a rotor ofthe motor by subtracting the rotor's actual speed form its referencespeed; generating a torque command from the generated speed error;generating an air gap power command by multiplying the generated torquecommand by the rotor's reference speed; generating stator resistivelosses from the generated torque command; generating an air gap error bysubtracting the estimated air gap power from the sum of the generatedair gap power command and the generated stator resistive losses; andgenerating the duty cycle command from the air gap error.
 23. A methodof generating a duty cycle command of a two-phase motor comprising twoseries-connected coil windings for each of the two phases, the methodcomprising: generating a first torque error for a first of the twowindings of a first of the two motor phases based on the current appliedto the first winding of the first phase and the position of a rotor ofthe motor; generating a second torque error for a second of the twowindings of the first motor phase based on the current applied to thesecond winding of the first phase and the rotor position; generating athird torque error for a first of the two windings of a second of thetwo motor phases based on the current applied to the first winding ofthe second phase and the rotor position; generating a fourth torqueerror for a second of the two windings of the second motor phase basedon the current applied to the second winding of the second phase and therotor position; generating, with a processor, a fifth torque error byadding the first, second, third, and fourth torque errors; andgenerating the duty cycle command from the fifth torque error and atorque error command.